Entropy of an autoequivalence on Calabi–Yau manifolds

We prove that the categorical entropy of the autoequivalence $\mathrm{T}_{\mathcal{O}} \circ (- \otimes \mathcal{O}(-1))$ on a Calabi–Yau manifold of dimension $d \geq 3$ is the unique positive real number $\lambda$ satisfying\[\sum_{k \geq1} \dfrac{\chi (\mathcal{O}(k))}{e^{k \lambda}} = e^{(d-1) t} \; \textrm{.}\]We then use this result to construct the first counterexamples of a conjecture on categorical entropy by Kikuta and Takahashi.

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I would like to thank Fabian Haiden for suggesting this problem to me, and Genki Ouchi for reading and pointing out Remark 4.3. I would also like to thank Philip Engel, Hansol Hong, Atsushi Kanazawa, Koji Shimizu, Yukinobu Toda, Cheng-Chiang Tsai and Chin-Lung Wang for helpful conversations and correspondences. Finally, I would like to thank Shing-Tung Yau and Harvard University Math Department for warm support.

Received 4 May 2017

Accepted 8 October 2017

Published 5 July 2018